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·.
Plastic Design of Unbraced Multistory Frames
SIMPLIFIED ANALYSIS OF UNBRACED FRAMES
by
Paul W. Reed
This research has been carried out as part of an investigation sponsored by the American Iron and Steel Institute
Department of Civil Engineering
Fritz Engineering Laboratory Lehigh University
Bethlehem, Pennsylvania
May 1972 Fritz Engineering Laboratory Report No. 367.9
ii
ACKNOWLEDGMENTS
This research was conducted at Fritz Engineering Laboratory,
Lehigh University, Bethlehem, Pa., and was carried out as part of an
investigation sponsored by the American Iron and Steel Institute.
Dr. Lynn S. Beedle is Director of Fritz Engineering Laboratory and
Dr. David A. VanHorn is Chairman of the Department of Civil Engineering.
The report was typed by Mrs. J. Neiser and the illustrations
were prepared by Mr. J. M. Gera and Mrs. S. Balogh.
1.
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7.
TABLE OF CONTENTS
ABSTRACT
INTRODUCTION
1.1 Preliminary Design
1.2 Purpose and Scope
SUBASSEMBLAGE METHOD OF ANALYSIS
2.1 The Assemblage
2.2 Sway Subassemblages
2.3 Restrained Column
2.4 Solution Procedure for Subassemblage Analysis
CHORD DRIFT IN THE SUBASSEMBLAGE ANALYSIS
3.1 Regions of the Frame Affected
3.2 Effect of Axial Shortening of Columns
NUMERICAL EXAMPLE
4.1 Comparisons to be Made Hith Other Methods
4.2 Results of Column Axial Shortening
4.3 Results of Subassemblage Analysis Compared to Other Methods
4.3.1 Lower and Middle Stories
4.3.2 Upper Stories
4.4 One-Story Assemblage Method of Analysis
SUMMARY AND CONCLUSIONS
NOMENCLATURE
TABLES
1
2
3
3
5
5
6
7
9
13
13
14
18
18
19
20
21
22
24
26
28
29
iii
iv
8. FIGURES 35
9. REFERENCES 50
.. I
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1
ABSTRACT
An analytical procedure is presented for determining approximate
elastic-plastic behavior of individual stories in an unbraced multi-story
steel frame subjected to nonproportional combined loading. The procedure
is based on sway subassemblages and considers the second order P-~ effect.
The original approach to this method is expanded to include column axial
shortening for analysis of individual stories. This is shown to affect
significantly horizontal drift of a story and is also shown to influence
the order of plastic hinge formation. The use of general parameters for
the assumptions of boundary conditions allows all regions of the frame to
be analysed. These parameters can be conservatively chosen, thereby
allowing a conservative analysis. Also, simplifying assumptions make the
method easy to apply to direct tabular computation.
The individual story behavior obtained by this method has been
compared with two full frame analyses, considered more exact. The
comparisons show very good agreement of results, indicating that this
approximate method is accurate and conservative. The results therefore
indicate the method is suitable for checking frame strength and stiffness.
•.
2
1. INTRODUCTION
The tall building will be a predominant structure for housing
people to live and work. Plastic methods may offer economy in the design
of steel buildings. Strength and stiffness criteria must be met to
achieve satisfactory preformance of tall buildings, and simplified methods
are needed to make certain these criteria are met. Therefore, this thesis
will present extensions to simplify and to make more general a method to
analyse unbraced frames called the sway subassemblage method of analysis.
The subassemblage method gives the utmost aid to the engineer to
visualize and compute the load deflection behavior of bare steel multistory
frames. Its conceptual contribution lies in its simplistic approach to
dividing the frame into smaller units which can be more easily handled
than trying to visualize the behavior of a highly redundant frame.
The use of the digital computer has immensely aided the analysis
of structures; however, often with "canned" programs, important structural
engineering concepts of analysis may be lost without proper scrutiny. On
the other hand, the subassemblage approach allo~vs close examination of a
small part of the frame, such as one story, which manifests the use of
this lucid approach to analysis. It can be applied either by hand
calculation or could be done efficiently by digital computer. It
achieves the non-proportional analysis of a story of a frame and
accomplishes the aim of an easy means of judging frame strength and
stiffness.
Because the subassemblage method relies upon the division of a
frame into small parts which are easily analysed, it is noticed that the
i
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3
benefit of this method is the potential for redesign once criteria have
been set for strength and stiffness. The potential to achieve an improved
design is important to the designer who is seeking economy and safety.
1.1 Preliminary Design
An extensive elaboration on the preliminary design of unbraced
frames is not presented in this thesis. Any method may be used to select
preliminary beam and column sections. Driscoll (8) and Hansell (9) present
plastic design procedures, using the technique of plastic moment balancing.
The moment balancing method bases the design on formulation of
equilibrium and plastic girder mechanism. The procedure for the design
of girders is reversed for analysis. Instead of calculating a required
plastic moment for selection of girder sections, the plastic moment
capacity is used to find maximum end moments for a beam to form a combined
load mechanism. From formulae (8 , 9) the maximum end moments for a
selected beam can be found. These moments are called "limiting moments"
and they are unique for each beam in that only these moments define a
girder mechanism. Later in this paper, these are used to find the
moment-rotation behavior of the beams of a sway subassemblage.
1.2 Purpose and Scope
It is the purpose of this thesis to improve the subassemblage
method of analysis, the approximate analytical method for predicting the
complete loadlieflection behavior of a story of an unbraced frame,
. . 11 d b D . 1 (J, 4 • S, G) or1g1na y presente y an1e s The modification to the
4
I
4
method emphasizes that accuracy is improved and that the method is made
more general, but that improvement will be within the scope of specific
approximations. The extended method is therefore not exact but is shoWn
to reasonably estimate strength and stiffness.
The original method is altered in two ways. First, the
assumption for the points of contraflexure of columns is not restricted
to midheight. This assumption is made general in order to handle most
regions of a frame. Second, the original approach made use of charts to
work the method by hand. Hand calculation is shown to be accomplished
directly by tabular computation without using design charts. Also,
Armacost(Z) applied the subassemblage method to digital computer by using
a small-step incremental approach. The direct computation worked for hand
calculation is applicable to digital computer without need of the
incremental approach.
The method is expanded to include the effect of axial shortening
in the columns. Total frame behavior influences the strength and
stiffness of each story through axial shortening. The inclusion of this
effect for unbraced frames is important and makes the method more reliable.
Emphasis is made on understanding the method and applying it to
engineering practice. Hence, this thesis solely intends to present
refinements to the subassemblage approach to improve its accuracy and to
give a reasonably easy method to apply.
. I
5
2. THE SUBASSEMBLAGE METHOD OF ANALYSIS
This chapter will make modification to the subassemblage method
of analysing frames as originally presented by Daniels. It will explain
the method and the assumptions used. A generalization of the method
enables it to be applied to most regions of a frame. The method will be
described from the point of view that simplifying assumptions make direct
tabular computation possible. Application to computer can allow the
method to be used in a more sophisticated manner without as many
simplifications.
2.1 The Assemblage
An assemblage is a model to represent the relationship between
horizontal shear resistance and sway deflection for a particular story of
a multistory frame. The assemblage consists of floor beams and a portion
of columns below the floor level extending down to assumed inflection
points as shown in Fig. 2.1. Different boundary conditions could be
appropriate for modeling top, middle, or bottom stories. Past analyses of
multistory frames(lO, ll) have shown that most middle and lower stories
had inflection points near midheight and actually most inflection points
were above midheight. The method of analysis in this thesis uses the
following assumption. Each column will have an inflection point at a
distance ah below the centerline of the floor girders. The value of a is
assumed to equal 0.5 for typical lower and middle stories.
. \
6
2.2 Sway Subassemblages
The assemblage is further separated into smaller models called
subassemblages. Each subassemblage consists of one column and either one
or two floor beams framing into the column top as shown in Fig. 2.2. The
far ends of the floor beams are assumed to sit on roller supports. Moment
resistance of the far ends of the beams is simulated by springs, idealizing
the beam to column restraint, and allowing beam end rotation. The sway
subassemblage is the simplest possible model for calculating the load
deflection behavior of a beam and column. Furthermore, the load-deflection
curves of each subassemblage are combined to get the complete load-
deflection behavior of the assemblage.
The influence of floor beams on the subassemblage is to provide
stiffness which restrains the rotation of the column. The restraining
moment on each beam is found using the following expression
= K EI9 L
2.1
Where K = the stiffness factor for the beam. In this thesis fhe following
assumptions are made concerning floor beams:
1. An elastic perfectly plastic moment-rotation relation is
~ assumed.
2. The girder stiffness K is assumed to equal 6.0 to work
the method by hand calculation. The stiffness is modified
(4) according to formulae by Danials to work the problem by
digital computer. If a plastic hinge forms at one end of
the girder, the stiffness is reduced to 3~0 at the other end.
._
7
3. Girder rotation B (clockwise rotation is taken as positive
on the end of the member) is assumed equal for each end
of the girders of a subassemblage. At formation of a
plastic hinge, the rotation is assumed to be unrestrained
for any further increment of rotation.
4. Beam axial shortening is neglected.
2.3 The Restrained Column
The column of a subassemblage is restrained by the floor beams
and is permitted to displace laterally. Each column shear Q is expressed
in an equilibrium equation for the freebody of the column shown in Fig. 2.3.
where M = u
p =
11 =
h =
a. =
Q M
u = a.h
upper end moment on the column
column thrust
story lateral displacement
story height
decimal portion of story height column to an assumed inflection
from the top of the point
Column end moments are determined from the equilibrium of
moments at a joint shown in Fig. 2.4. The sum of column end moments
2.2
above and below the joint equals the sum of beam end moments at the joint
called restraining moment M
M r
=
r.
2.3
where MBL = girder end moment lert of joint
MBR girder end moment right of joint
The column end moment M is assumed to be a portion ~ of the restraining u
moment.
8
M = 13 M 2.4 u r
The value of 13 is assumed to equal - 0.5 for typical middle and lower
stories. Daniels showed this assumption to be conservative. For analysing
other than middle and lower stories, another assumption for 13 could be made.
The angles for the free body diagram in Fig. 2.3 of the
restrained column have the compatibility relationship:
/:;,.
h g·- y
where 9 = rotation of the joint
y = the angle between the chord of the column segment and the tangent to the column centerline at the joint.
The rotation of the restrained column y is a function of the
2.5
column moment M and the axial thrust P. A method to relate the momentu
rotation-thrust of the column required use of charts by Daniels. Armacost
approximated this relationship by fitting an equation to the curves prepared
by Daniels by assuming the initial slope of the curves closely approximated
the ascending portion of the curves. These expressions have been
considerably improved and have been made appropriate for strong and weak
axis bending of wide flange sections by the following:
·.
•.
where
or
M u
y = J X M pc
J __ [arhx P ] -5 ( 25-22 py ) - 72 X 10
for strong axis bending
for weak axis bending
p -- + p
y
The thrust in each column is assumed to be constant for
combined load analysis, as was done by Daniels. A preliminarv design
9
2.6
approach is used to find these column thrusts. A conservative estimate is
made by assuming the girder end moment sum caused bv lateral force to be
distributed to each girder in proportion to their limiting end moment sum.
The vertical end shears in the girders are thus found for lateral load bv
dividing the girder end moment sum by its length. These vertical shears
are summed from the top floor dmmward and are added to the column thrusts
based on gravity tributary area at each floor level to get the column
thrusts for combined loading.
2.4 Solution Procedure for Subassemblage Analysis
The procedure to find the load-deflection behavior of an
assemblage is next described. It is remembered that this method is a
non-proportional displacement method because distributed gravity load is
applied first and subsequent lateral load is found for a certain
displacement. This method is not incremental as was nresented hv
·.
·-
10
Armacost. The procedure is summarized below in four steps:
1. Determine the initial moments on the beams under factored
distributed gravity load. A one-story moment distribution
is used to find the end moments under no lateral load. The
limiting moments for the floor beams are found as described
in Chapter 1. Each column limiting moment equals the reduced
plastic moment under its factored axial force. The initial
values of shear resistance Q are found for each column from
Eq. 2.2 under no sway deflection.
2. Each subassemblage is analysed separately to find its load
deflection behavior. This step consists of determining the
sequence of plastic hinge formation as each increment of
rotation changes beam end moments from the initial state
under vertical load to the final combined load state. The
initial beam end moments are subtracted from the corresponding
"limiting moments" to find the possible change in moment to
form a plastic hinge. The amount of relative rotation 9
necessary to cause this change is found by using Eq. 2.1.
The minimum rotation for all beam ends controls and the
controlling rotation is used to find the change of moments
in the beams. These changes of beam moments are then added
to the previous state to determine the new moments on the
subassemblage. The restrained column end moment is found
using Eq. 2.4. After formation of a plastic hinge at some
point in the subassemblage, the stiffness is reduced to zero
·.
11
at that point. This is the same as inserting a real hinge.
A change of stiffness at other locations of the subassemblage
may be necessary after a plastic hinge has formed. For
example, if a plastic hinge forms at one end of a beam, the
stiffness at the other end is reduced to K=3.0. The process
of finding new moments and rotations for each successive
plastic hinge is continued up to formation of a subassemblage
mechanism. The rotations and column moments are saved and
used in the next step.
3. The shear resistance and drift index are next calculated at
formation of each plastic hinge. The drfft index is found
using Eq. 2.5 where 9 is known from (2) and_y is found using
Eq. 2.6. Finally, the subassemblage shear resistance is
calculated from Eq. 2.2, using the values of column moment
from (2) and ~/h from (3). The monotonic relationship of
horizontal shear versus drift is available and shows the
complete load-deflection response of the subassemblage.
4. The monotonic load versus drift curves are combined for all
subassemblages to obtain the load-deflection curve of the
assemblage.
It is noticed from the equation for equilibrium of the restrained
column, Eq. 2.2, that the value of subassemblage shear resistance is
conservative for an assumed point of contraflexure lower than the true
point. From Eq. 2.2 the critical parameter is the value ofa.. For a.·
greater than the true a, a calculated value of shear will be less than
the true shear resistance. Thus, the calculated value for strength
would be on the safe side. The choice of this parameter becomes very
important in determining whether the analysis is higher or lower than
the true solution.
12
To consider fixed base bottom story columns, the parameter a is
also important. Under no lateral load, the point of contraflexure of a
bottom story column may exist below midheight. Application of lateral
load results in the point of contraflexure climbing towards the column top.
Finally, the column may be bent in single curvature. For single curvature,
an imaginary point of inflection actually lies above the column top. The
combined load analysis presented in this thesis for a bottom story would
be poor for a large variation of inflection point. But, a reasonable value
of assumed a can give a safe analysis as long as it is assured that the
true a is less than the assumed a. Then, even the case of single
curvature bending would result in a conservative analysis.
For most middle and lower stories the assumption that a=0.5 is
conservative. This would lead to a conservative horizontal shear
resistance but it remains to be seen whether sway deflection estimates
are on the safe side. This thesis intends to show that sway deflection is
affected by the influence of column axial shortening and previous uses
of this method were unconservative as a drift estimate.
·.
•.
13
3. CHORD DRIFT IN THE SUBASSEMBLAGE ANALYSIS
This chapter will extend the analysis of the assemblage to
include the effect of column axial shortening. The assumptions to separate
the assemblage from the frame made the analysis very simple. They clearly
emphasize conservatism in calculating the strength of the assemblage.
First, the assumption that an inflection point is below the true inflection
point can be conservatively made, and second, the assumption of the end
moment in the restrained column is conservative by taking a safe proportion
of the restraining moment M . The use of these assumptions also makes r
a sway deflection estimate conservative for only the assemblage but if the
action of the frame below the assemblage is considered, the resultant sway
deflection will increase. Also, additional moments are caused by
differential column axial strain. This chapter describes an approximate
method to include the effect of chord drift in the subassemblage analysis.
3.1 Regions of the Frame Affected
The major influence of chord drift would be on relatively tall
frames and mainly in the middle and upper regions of multistory frames.
As pointed out by Kim(lO), the influence of chord drift was minimal under
nonproportional loading for low frames but was significant in a 26 story
frame. Also, Parikh(ll) demonstrated its effect to be significant for a
24 story frame. The height to width ratio for which chord drift would
have importance would be difficult to find, therefore, each frame should
be checked individually to assign relative importance to this effect.
It is proposed that chord drift should be included in the
analysis of frames of relatively high vertical dimension. It is
described herein how to include the effects of chord drift in the sway
subassemblage method of analysis.
3.2 Effect of Axial Shortening of Columns
14
The action of column axial shortening affects the strength and
stiffness of an assemblage in two ways. First, the differential shortening
of columns causes drift in an assemblage due to a geometric change of the
frame below the story being considered. As shown in Fig. 3.1, the
exaggerated shortening of column EF has caused the top of the frame BCDE
to deflect horizontally relative to story AREF. This horizontal deflection
is caused by a geometric change in the frame. With a much larger and
complex frame, differential shortening will not be as easily scrutinized
as in this two story example but will vary from column to column.
To account for sway deflection in an assemblage caused by this
geometric effect, a virtual work method is used. It is assumed that the
longitudinal strain in the columns is elastic under all loading cases.
The foundation of the virtual work method lies in establishing an
equilibrium system for the structure under unit loading. A deforrnation
system, resulting from actual loading, is superimposed onto the
equilibrium system to find the defection of the frame.
The actual axial load in the columns is constantly varying
under proportional application of lateral load. For simplicity, it is
assumed that the axial force in the columns is constant under combined
gravity plus lateral load. The column thrust for combined load is found,
15
using the concept in Chapter 2, by distributing the moments in a story
caused by lateral load in proportion to their limiting end moments. The
resulting column thrusts P are achieved only for combined load mechanism,
and they are considered to be a conservative set of forces. Then, the
actual column strain equals
= p
AE 3.1
The equilibrium system is formulated by applying a unit lateral
load to the floor level under investigation and a unit lateral load is
applied to the next lower floor level in the opposite direction, shown in
Fig. 3.2. By applying the unit loads in this manner, a deflection of only
one story is found. This deflection is a relative deflection of one floor
level to the next lower floor level. The column axial forces due to the
unit load system are found using an approximate method developed by
Spurr(l2). The axial force in a column is taken as proportional to the
relative column areas and the distance from the column group centroidal
axis. This axial force N of the equilibrium system in one line of
columns is assumed equal for all columns. Thereby, the relative horizontal
story deflection ~ is a summation of axial force due to unit load times
the actual elastic strain of each column up to the floor level that
deflection is required as expressed by the following:
3.2
This relative deflection due to column axial shortening has been
determined for the factored combined loading case. To use this deflection
at working load, it is assumed that the deflection due to axial shortening
16
is divided by the load factor (F=l.3). Furthermore, after reaching the
factored lateral load, the deflection due to axial shortening is assumed
constant. This deflection can be used in the column equilibrium equation
Eq. 2.2 under the above assumptions. Relative magnitudes of Eq. 2.2 show
that shear resistance is not significantly reduced when this effect is
included.
The second part for the inclusion of axial shortening in the
subassemblage method of analysis is described herein. As shown in
Fig. 3.3, if CD settles a distance 6 , the frame ABCD is subjected to sway v
and the members are subjected to bending beyond any loading condition.
This is similar to column shortening in a multistory frame. Application
of lateral load causes differential joint displacement, resulting from
column shortening, and this joint displacement causes additional moment.
Under combined loading, the previously mentioned assumption
of constant axial force results in constant joint settlement due to column
axial shortening. The amount of differential settlement 6, of joints J
gives fixed end moments from the equation
F~ 6EI
= ~
6. J
For each assemblage, a one-story moment distribution will give a final
moment diagram for the moments caused by axial shortening. To include
these moments in the subassemblage analysis, it is assumed that these
moments are subtracted from the limiting moments to give new "limiting
end moments". These new limiting moments would be used in the same
procedure for executing the analysis as described in Chapter 2.
3.3
17
To demonstrate the use of the subassemblage analysis as
described in Chapter 2 and to show the effects of column axial shortening,
an illustrative example is provided in the following chapter.
18
4. NUMERICAL EXAMPLE
The sway subassemblage method described in the preceeding
chapters is next used to analyse an example frame. Several load deflection
curves of one-story assemblages will be presented for comparison with
corresponding story behavior calculated by other methods.
The frame shown in Fig. 4.1 uses the geometry and loading of
the design example given in Ref. 1. The original example showed the
plastic design of a braced frame; the example in this thesis shows a new
(9) design made by plastic procedures of an unbraced frame. It is a three-
bay, twenty-four story frame with distributed loads using steel with a
36 ksi stress level, and it is designed such that the bare steel skeleton
is required to carry. all loadings. The beams were designed using clear
span length and live load reduction was considered for both beams and
columns. The design ultimate load is equal to 1.3 times the working
load, corresponding to the combined loading condition, and is equal to
1.7 times the working load, corresponding to the gravity loading case.
The design of this frame is preliminary, and as such, secondary checks have
as yet not been made. The member sizes are shown in Tables 4.1 and 4.2
and the gravity loads are shown in Tab. 4.3. The design is used to check ·.
the suitability of the subassemblage method to analyse frames.
·-
4.1 Comparisons To Be Made With Other Methods
To demonstrate effectively the use of the assemblage for
evaluation of frame strength and drift, the subassemblage method is compared
. (10) to the sway 1ncrement method and Parikh's elastic-plastic analysis of
19
unbraced frames(ll); both developed at Lehigh University. The sway
increment method provides for the entire frame a nonproportional loading
analysis which gives the lateral shear resistance of the frame and sway
deflection of stories as plastic hinge formation progresses to the point
of frame failure. Parikh's analysis provides a proportional loading
analysis of the entire fram~ and in this thesis it is used to check the
working load deflections of individual stories.
The comparisons of these analyses will consider strength and
story drift as the main criteria for adequately judging frame behavior.
Working load deflection is the main concern for judging the story drift
but this criterion is limited because the largest permissible drift index
is uncertain. In this thesis any reasonable drift index will be considered
to be acceptable. The criterion for strength for combined load analysis
is the amount of shear resistance available in the frame. The shear
resistance for an assemblage should be greater than the design ultimate
shear for each floor level. To be conservative the shear resistance
found by the subassemblage method should be less than that found by overall
frame analysis.
4.2 Results of Column Axial Shortening
Inclusion of column axial shortening in the subassemblage
analysis has been shown to be significant in earlier references. To
demonstrate the effect of chord drift using the subassemblage method of
analysis, floor level 14 of the example frame is analysed both neglecting
and including this effect. In Fig. 4.2 the load-deflection curves are
20
shown, in which the load Q, plotted on the vertical axis, is the shear
resistance of the assemblage and the drift index, plotted on the horizontal
axis, is the horizontal sway deflection of a story divided by the story
height below the floor level. The two curves are very similar in shape.
The inclusion of chord drift shows an increase in sway deflection between
20 and 30% which is substantial and therefore should not be neglected.
The maximum shear resistances for the two curves are nearly the same. The
strength of the assemblage decreases only slightly when chord drift is
included and thus overall strength is not significantly affected. The
successive formation of plastic hinges for each subassemblage is shown in
Fig. 4.3. No change in plastic hinge formation occurs in subassemlilages
A or D. In subassemblages B and C, it is noticed that inclusion of chord
drift, caused the exterior beams of the two subassemblages rather than
the interior beam to form plastic hinges first. This results from the
decrease in positive beam end moment of the interior beam due to differential
shortening of the interior columns; thus it took much more positive rotation
Q to form plastic hinges in the interior beam. With increased positive
rotation the exterior beams have plastic hinges form first. Hence, the
effect of column differential shortening on an assemblage is an increase
in horizontal sway deflection and overstressing occurs at different
locations. Similar results were obtained for other floor levels.
4.3 Results of Subassemblage Analysis Compared to Other Methods
Next, the load-deflection curves for several different floor
levels are compared using the simplified subassemblage approach and using
the two frame analyses described earlier. For convenience, the story
behaviors for these analyses are compared in three parts of the frame:
the lower, middle and upper stories.
4.3.1 Middle and Lower Stories
21
In Fig. 4.4 the load-deflection curve of floor level 19 compares
the subassemblage method to the sway increment method of overall frame
behavior and Parikh's working load deflection. All analyses include the
effect of column differential shortening. For the subassemblage method,
the values of a and S of Eqs. 2.2 and 2.4 are assumed to equal 0.5.
Excellent agreement is evident. The sway increment method shows larger
shear resistance up to the point where the frame lost stiffness and the
curve is discontinued, telling nothing further about the story behavior.
The subassemblage analysis yields a lower shear strength as expected from
the conservative assumptions. The subassemblage approach also provides
the complete curve through unloading of the lateral force, and thus shows
how the story behaves when separated from the frame. The working load
deflection is the same for both methods and is slightly less than the drift
index given by Parikh's analysis.
The load-deflection curves for levels 17 and 14 are shown in
Fig. 4.5 and 4.6, respectively. Both curves of the subassemblage approach
show good agreement with the sway increment method. The maximum lateral
load is less for the subassemblage approach which again shows expected
conservatism in the assumptions. The subassemblage method still indicates
strength greater than the design ultimate loa~ showing that the story
legitimately satisfies the strength criterion. The working load sway
·-
·-
22
deflection of level 17 for the suhassemblage approach is the same as for
Parikh's analysis although less than for the sway increment method. For
level 14 the working load sway deflection is slightly greater for the
subassemblage method than for Parikh's analysis. These results show
that story deflection is very closely approximated by the subassemblage
approach.
The load-deflection curves for floor level 10 are shown in
Fig. 4.7. The value of Sis assumed to equal -0.6 for this level. The
subassemblage analysis is conservative ir. showing less strength than the
sway increment analysis and also is conservative in showing a greater
working load deflection than either Parikh's analysis or the sway increment
method. The slight disagreement of results at this level is partly due to
the conservative assumption of inflection point at midheight. Also, the
inclusion of increased sway caused by chord drift is conservative.
4.3.2 Upper Stories
The design of upper stories is generally controlled by the
gravity loading case and, as such, combined load analysis may not be
necessary. However, the transition from the gravity load controlled
region to the combined load controlled region may not be readily apparent.
To better analyse the upper stories for the combined loading case, different
assumptions for a and S may be necessary.
The load-deflection curve for floor level 6 is shown in Fig. 4.8.
For the subassemblage approach, the value of a was assumed to equal 0.75
and the value of S was assumed to equal -0.5. The subassemblage analvsis
shows a slightly larger shear resistance and it is in very good agreement
23
with the sway increment method. The working load deflection is the
same as Parikh's analysis. The assumption of a proves to be acceptable
in giving a good subassemblage analysis for this region of the frame.
The load-deflection curve for floor level 2 is shown in
Fig. 4.9. The value of a is assumed to equal 0.75 which is the same as
for level 6. The initial part of the curve agrees exactly with the sway
increment analysis and the working load drift index is the same as Parikh·'s· . {·•
analysis. The maximum lateral load of the subassemblage approach is much
larger than from calculations in the sway increment method. The sway
increment method stopped earlier due to frame failure at another location
in the frame, resulting in a lack of information of the complete behavior
at this level. Thus, the subassemblage method gives a complete curve,
although it is uncertain if the true behavior is given beyond the results
of the sway increment method. However, so near the top of the frame, the
combined load analysis need not be exact because this region is undoubtably
controlled by gravity loading.
A comparison of column axial thrusts is made at factored combined
gravity plus lateral load in Tab. 4.3 for the several analyses of floor
levels 2, 5, 10, 14, 17, 19. The column thrusts were computed by Parikh's
·. analysis and by the estimate for subassemblage analysis given in Chapter 2.
•. By the subassemblage analysis these thrusts were considered constant. The
thrusts from both methods are reasonably close. The thrusts in the leeward
interior and exterior columns are greater for the subassemblage analysis.
The total sum of column thrusts being constant for each floor level results
in the windward interior and exterior columns carrying less thrust for the
24
subassemblage method. The leeward columns therefore will show conservatism
when axial thrust is considered in the subassemblage approach.
A comparison of the cumulative shortening of the columns below
a certain level is made in Tab. 4.4 for factored combined gravity plus
lateral load. The two methods compared are the Parikh analysis and the
estimate for the subassemblage approach. The downward joint displacements
are very close to one another because the axial forces in columns are
close by both methods. This indicates that the differential joint
displacements will be fairly accurate in finding fixed ended moments due
to column axial .shortening.
4.4 One Story Assemblage Method of Analysis
Along with the sway increment analysis, a one story assemblage
h d d 1 d b K. (10) met o was eve ope y 1m . This approach has a similar assumption
to the subassemblage approach; it assumes the location of inflection point
at midheight of the columns. This is more restrictive than the
subassemblage approach presented in this paper although it has been shown
to be acceptable for middle and lower regions of the frame. (4) A
comparison of the one-story assemblage analysis to the subassemblage
analysis is shown in Fig. 4.10. where the load-deflection curves of
floor level 14 are plotted. Previously, the maximum lateral load for the
subassemblage analysis was generally less than for the entire frame
analysis. The one-story assemblage analysis shows a greater reduction
in predicted shear strength and is therefore more conservative than the
subassemblage analysis. Analyses of other middle and lower levels also
25
indicate greater conservatism in this approach.
4.5 Summary
In this comparative study, it is noticed that the subassemblage
analysis showed very good agreement with more exact full frame analyses.
This analysis is crude due to simplifying assumptions and, as such, it
should not be expected to be exact. The results of the comparisons show
that the extended but simplified method shows conservatism for both strength
and sway deflection. It, therefore, proves that if a floor level analysed
by this approach is found to be satisfactor» then the level will be
acceptable in context with the entire frame.
\
26
5. SUMMARY AND CONCLUSIONS
It has been the purpose of this thesis to extend the
subassemblage method of analysing unbraced frames. The analysis is used
to predict elastic-plastic second order behavior under non proportional
loading. A general procedure is made by including several effects not
included in the original development of the method. The possible use
of different than midheight points of inflection for separating an
assemblage from the frame was developed. This could allow use of the
subassemblage method for upper stories or bottom story as well as for
middle and lower regions which use the midheight as the inflection
point. The effect of column axial shortening was extended to the
subassemblage analysis. The sway subassemblage method is considerably
simplified with the use of a direct computation for finding the minimum
rotation at formation of each successive plastic hinge. This is applicable
to hand computation or digital computer usage replacing the highly
inefficient incrementation of rotation by very small amounts as suggested
in Ref. 2 to find the complete load-deflection curve of an assemblage.
From the analytical results of a design example given for an
unbraced multistory frame, it can be concluded that:
1. The effect of column axial shortening is not considerable
in changing the lateral load capacity of an assemblage. The
major effect is an increase of lateral deflection, and over
stressing at different parts of the assemblage causes a change
in the order of plastic hinge formation.
. '
27
2. A choice of point of inflection at midheight makes the
subassemblage approach conservative for middle and lower
stories • For other than lower and middle stories, a choice
of a below midheight helps to make the analysis more reliable.
Guaranteed conservatism is possible for a choice of inflection
point below the actual point of inflection.
3. A comparison with more exact overall frame analyses shows
that the extended subassemblage approach is conservative in
its estimation of both lateral load capacity and horizontal
sway deflection. Therefore, the simplified approach can be
used to evaluate the behavior of unbraced frames.
A
E
FEM
h
J
I
L
MB
M pc
M u
p
p y
Q
rx' r y
L:
a
.._
6. NOMENCLATURE
Area of wide flange sections;
Modulus of Elasticity;
Fixed end moment;
Story height;
Term relating moment-rotation-thrust of a column;
Moment of inertia;
Span length;
Beam enci moment;
Reduced plastic moment;
Column end moment;
Column thrust;
Column axial yield load;
Horizontal shear force on a column;
Column radius of gyration, x-axis and y-axis;
Finite summation;
Decimal portion of story height from fue column top to an assumed inflection point;
Decimal portion of the sum of beam end moments at a joint, assumed to equal the column top moment;
Column chord rotation;
Horizontal displacement of the column top relative to the column bottom;
Axial column strain;
Rotation of the joint.
28
29
7. TABLES
·-
Table 4.1 Beam Sections for Example Frame 30
Level AB and CD BC
1 Wl6x31 Wl0x21
2 Wl6x31 Wl2x22
3 .. Wl6x36 Wl2x22
4 Wl6x36 Wl4x22
5 Wl6x36 Wl4x26
6 Wl6x36 Wl4x26
7 Wl6x40 Wl4x26
8 W16x40 W14x26
9 W16x40 W14x26
10 W18x40 Wl6x26
11 W18x40 W16x26
12 W18x45 W18x35
13 Wl8x45 W18x35
14 W18x45 H18x35
15 H21x44 Wl8x40
16 W21x44 Hl8x40
17 W21x44 H18x40
18 W18x55 Hl8x55
19 Wl8x55 Hl8x55
20 H18x55 W18x55
21 W21x49 W2lx49
22 W2lx49 W21x49
23 W21x55 H21x55
24 W2lx68 W21x68
Table 4.2 Column Sections for Example Frame 31
Level A and D B and C
1 Wl2x40 W12x40
2 Wl2x40 W12x40
3 Wl2x40 W12x40
4 W12x40 W12x40
5 W12x58 W12x58
6 H12x58 W12x58
7 t\112x79 W12x79
8 W12x79 W12x79
9 W12x85 H12x85
10 W12x85 H12x85
11 W14xlll W14x111
12 W14xll1 W14xll1
13 W14x111 H14xll9
14 W14x111 W14xll9
15 W14x127 H14x136
16 W14x127 W14xl36
17 H14x150 W14xl58
18 H14x150 W14x158
19 W14x158 W14x176
20 W14x158 W14x176
21 W14x176 H14x193
22 Wl4x176 Wl4x193
23 W14x219 Wl4x237
24 t.Jl4x219 Wl4x237
Table 4.3 Working Gravity Loads For Example Frame 32
(a) Beam Loads (K/ft)
Level AB and CD BC
1 1.80 1. 8()
2-24 1. 78 3.03
(b) Joint Loads (kip)
Level A and D B and c 1 8.63 2.43
2 20.39 3.05
3-22 15.97 -1.19*
23-24 16.46 -0.89*
*Due to live load reduction in columns
--
--
Table 4.4 Axial Loads on Columns for Factored Combined Loading
Level Column Axial Load Below Level A
Col. A Col. B Col. c Col. D (kips)
2 97.4 114.4 102.8 103.1
6 282.2 324.9 308.2 319.5
10 459.6 519.7 540.2 556.0
14 626.3 684.1 804.9 803.2
17 745.9 786.7 1023.7 995.7
19 820.7 837.1 1188.3 1127.0
Level Column Axial Load Below Level B
Col. A Col. B Col. C Col. D (kips)
2 98.2 105.7 110.8 103.1
6 280.8 298.3 331.7 325.2
10 450.5 490.0 565.9 571.6
14 604.4 657.2 826.8 833.6
17 711.4 760.3 1043.7 1040.6
19 778.9 810.8 1207.2 1181.1
A Parikh's Analysis
B Estimate for Subassemblage Analysis
33
--
Table 4.5 Cumulative Shortening of Columns for Combined Loading
Level Displacement Below LevelA
Jt.l Jt.2 Jt.3 Jt.4 (inches)
2 1.53 1.57 1.89 1. 95
6 1. 33 1.34 1.68 1. 73
10 1.07 1.05 1.4 1.43
14 0.8 0.75 1.07 1.10
17 0.58 0.52 0.80 0.81
19 0.44 0.39 0.62 0.62
Level Displacement Below LevelB
Jt.l Jt. 2 Jt.3 Jt.4 (inches)
\ 2 1.49 1.51 1. 97 2.03
6 1. 27 1.28 1. 73 1. 80
10 1.02 1.01 1.43 1.50
14 0.76 0.73 1.09 1.15 . 17 0.54 0.51 0.81 0.85
19 0.41 0.38 0.62 0.65
A Parikh's Analysis
B Estimate for Subassemblage Analysis
34
35
8. FIGURES
--
•' :.
p p (n-I)A (n-l)s
p (n-1) c
M(n-I)A~ M(n-l)s
W AB
Hn---
~ig. 2.1 nne-Storv Assemblage
p (n-l)o
CX:h
l7
CX:h
I I I I I I I
·-
F_i_g. 2. 2 Typical Suhassemblages
O:h
-.
p
Q
~Mu
j __
p
Fig. 2.3 qestrained Column
p
•
p
ML(n-1)
) MeR
Mu( n)
F·;0 ).4 Eoui.lihrjum of ;'1ornents at Jnint
38
c
8
A
·-
Fig. 3.1
7/T
D
F
Column Axial Shortening Causes a Frame Geometry Change
Lev. n
Lev. n-1
7/T
Fig. 3.2 Application of Dummy Unit Load System
8
A
c
D
!J.v
Fig. 3.3 Settlement at D Causes Frame Change
39
·-
Level R
2
3
4
5
6
19
20
21
22
23
24
25 A
40
-
22@ 9'- 8 11
- 1-
121
- t-
i2 1
7/T B;y Ch:r Dp-._
1- 271
Fig. 4.1 Example Frame
Q (kip)
100 (
I I
,...-- ---- -- -- ----::~~~~-~-~-----~
With } Axial Shortening Without of Columns
0.01 0.02
Fig. 4.2 LoRd-Deflection Curve of Level 14
41
A
B
c
D
A
B
c . . ~
D
• Plastic Hinge
Without Chord Drift
0 • 2
:r •
rl 0 • 3 2
~
~I i th Chord Drift
• 2
r3 • 2
~I • • 3 I
~ Fig. 4.3 Order of Plastic Hinge Formation
for Level 14
42
150.
Q 100 (kip}
0
/ /
/I /
/
__ D.L.
...,._ __ w. L.
One - Story Su bassemblage Analysis
---Sway Increment Analysis of Frame, Ref. 10
• Proportiona I Elastic - Plastic Analysis, Ref. II
0.01
~/h
0.02
Fig. 4.4 Load-Deflection Curve of Level 19
43
Q (kip)
100
0
I I I I I
I' I
,~
0.01
~/h
Fig. 4.5 Load-Deflection Curve of Level 17
44 :. ·" ... ·-·
0.02
Q
(kip)
100
0
I /
/
/ /
, / ,
I D.L.
W.L ..
0.01
~/h.
Fig. 4.6 Load-Deflection Curve of Level 14
45
0.02
Q (kip)
100
50
Fig. 4.7
46
//
/ I
I
I I
0.02
Load-Deflection.Curve of Level 10
I
... ~·
Q
(kip)
75
50
25
0
W. L.
0.01
~lh
Fig. 4.8 Load-Deflection Curve of Level 6
47
0.02
4R
75
Q 50 (kip)
0.02
,
Fig. 4.9 Load-Deflection Curve of Level 2
•
•
Q (kip)
50
0
r-----1-------~--:..,c._- D. L.
--One- Story Subassemblage Analysis
--- One- Story Assemblage
Analysis, Ref. 10
0.02
Fig. 4.10 Load-Deflection Curve of Level 14
I . ;
y
""
9 . REFERENCES
1. American Iron and Steel Institute PLASTIC DESIGN OF BRACED MULTISTORY StEEL PFAMES,
New York, 1968.
2. Armacost, J. O.,III and Driscoll, G. C., Jr. THE COMPUTER ANALYSIS OF UNBRACED MULTI-STORY FRAMES,
Fritz Engineering Laboratory Report No. 345.5, Lehigh University, June 1968.
3. Daniels, J. H. and Lu, L. W. DESIGN CHARTS FOR THE SUBASSEMBLAGE METHOD OF DESIGNING
UNBRACED MULTI-STORY FRAMES, Fritz Engineering Laboratory Report No. 273.54, Lehigh University, March 1966.
4. Daniels, J. H.
50
COMBINED LOAD ANALYSIS OF UNBRACED FRAMES, Ph.D. Dissertation, Lehigh University, 1967, University Microfilms, Inc., Ann Arbor, Michigan
5. Daniels, J. H. A PLASTIC METHOD FOR UNBRACED FRAME DESIGN, American
Institute of Steel Construction Engineering Journal, Vol. 3, No. 4, October 1966.
6. Daniels, J. H. and Lu, L. W. SWAY SUBASSEMBLAGES FOR UNBRACED FRAMES, ASCE Meeting,
Preprint 717, October 1968.
7. Driscoll, G. c., Jr., Armacost, J. 0., III and Hansell, W. C. PLASTIC DESIGN OF MULTI-STORY FRAMES BY COMPUTER, Journal
of the Structural Division, ASCE, Vol. 93, STl, January 1970.
8. Driscoll, G. C., Jr., et al
9.
·10.
PLASTIC DESIGN OF MULTI-STORY FRAMES, Fritz Engineering Laboratory Report No. 273.20, August 1965.
Hansell, W. C . PRELUIINARY DESIGN OF UNBRACED MULTI-STORY FRAMES, Ph.D.
Di~sertation, Lehigh University, 1966, University Microfilms, Inc. , Ann Arbor, Mi·chigan.
Kim, J. W. and Daniels, J. H. ELASTIC-PLASTIC ANALYSIS OF UNBRACED FRAMES, Fritz Engineering
Laboratory Report No. 346.5, Lehigh University, March 1971.
, .
11. Parikh, B. P. ELASTIC-PLASTIC ANALYSIS AND DESIGN OF UNBRACED MULTI-STORY
STEEL FRAMES, Ph.D. Dissertation, Lehigh University, 1966, University Microfilms, Inc., Ann Arbor, Michigan.
12. Spurr WIND BRACING, McGraw-Hill, New York, 1930.
51